Statistical Natural Language Processing in R

Bennett Kleinberg

The bigger picture of textual data

Why textual data?

  • textual data are everywhere
  • textual data are (very) promising

Two general approaches

  1. Measurement
  2. Prediction

One big problem

  • text data \(\neq\) numerical data
  • compare this to survey data, trading data, crime statistics, etc.
  • a piece of text data is just that, “a text”
  • but: for statistical analyses (and statistical learning), we rely on numerical representations

The first challenge of NLP is text quantification.

This course

An introduction to statistical natural language processing in R

  • the basics of handling textual data in R
  • fundamental approaches of (real) NLP
  • text representations
  • measuring constructs in textual data
  • textual data for prediction (supervised machine learning)

Structure

  • short theoretical parts (4x)
  • followed by practicals (4x)
  • discussion of solutions

All materials: https://github.com/ben-aaron188/snlp

Ask all your questions

The quanteda package

library(quanteda)
packageVersion("quanteda")
## [1] '4.1.0'

Aspects of textual data

How would you describe this text?

The whole situation makes me very anxious. I do not believe, as a country, we are doing everything to prevent the spread. I believe we should be in a full and proper lockdown like other countries. I fear for my children and their futures.

Different aspects of text data

  • lexical dimension (text metrics)
  • syntactic dimension
  • semantic dimension

Lexical dimension

Numerical summaries of units of a document or collection of documents.

Examples:

  • no. of words
  • average sentence length

Syntactic dimension

Information about the arrangement/structure of language.

Examples:

  • grammatical function of words: verbs, nouns (parts-of-speech)
  • named entities: persons, locations, organisations
  • structure of a sentence (parse tree)

POS tags in Penn Treebank

Example of named entity tags (Spacy)

Textwash example

Semantic dimension

Information of the text concerned with meaning.

  • sentiment
  • psycholinguistic features
  • relationship between words
  • contextual variation of words
  • higher-order constructs

Units of textual data

What can we work with?

  1. A corpus is a collection of documents
  2. A document is a collection of sentences
  3. A sentence consists of words
  4. A word consists of syllables
  5. A syllable consists of characters

Special units in NLP

  • tokens: a sub-unit of text that is the at the core of NLP
  • ngrams: a sequence of \(n\) tokens

Example sentence:

The whole situation makes me very anxious.

Tokenisation

text_1 = 'The whole situation makes me very anxious.'

toks = tokens(text_1)
toks
## Tokens consisting of 1 document.
## text1 :
## [1] "The"       "whole"     "situation" "makes"     "me"        "very"     
## [7] "anxious"   "."

Common preprocessing steps

Removing punctuation:

toks_no_punct = tokens(text_1
              , remove_punct = T
              , remove_numbers = F
              , remove_url = F
              , remove_separators = F)
toks_no_punct
## Tokens consisting of 1 document.
## text1 :
##  [1] "The"       " "         "whole"     " "         "situation" " "        
##  [7] "makes"     " "         "me"        " "         "very"      " "        
## [ ... and 1 more ]

Note on tokenisation

This is far from a solved problem:

  • many tokeniser algorithms exist
  • each has a slightly different token segmentation approach
text_2 = 'evidence-based self-fulfilling prophesy'

tokens(text_2, what = 'word')
## Tokens consisting of 1 document.
## text1 :
## [1] "evidence-based"  "self-fulfilling" "prophesy"

Slightly different tokenisation

tokens(text_2, split_hyphens = T)
## Tokens consisting of 1 document.
## text1 :
## [1] "evidence"   "-"          "based"      "self"       "-"         
## [6] "fulfilling" "prophesy"

Other text units

# sentences
length(tokens(toks, what = 'sentence'))
## [1] 1
# tokens
ntoken(toks)
## text1 
##     8
# characters
nchar(text_1)
## [1] 42

Text metrics (corpus statistics)

Key idea

  • summarise key information of text data numerically
  • largely pertains to meta features of documents

Common text metrics: readability (see quanteda.textstats), lexical diversity

Take two examples

Example 1:

I am very anxious about this situation. Very anxious.

Example 2:

I am very anxious about this situation. Really scary.

Which has a higher lexical diversity?

Types and tokens

ex_1 = 'I am very anxious about this situation. Very anxious.'
ex_2 = 'I am very anxious about this situation. Really scary.'
  • Tokens: a sub-unit of text
  • Types: unique tokens

The type-token ration (TTR) is often used as a measure of lexical diversity.

TTR for our examples

ttr_1 = ntype(tokens(ex_1))/ntoken(tokens(ex_1))
ttr_1
##     text1 
## 0.8181818
ttr_2 = ntype(tokens(ex_2))/ntoken(tokens(ex_2))
ttr_2
##     text1 
## 0.9090909

Handling textual data in R

The stringr package

  • most powerful tool for most text string queries
  • based on regex (regular expressions)
  • useful cheat sheet

The corpus object

A collection of documents is a corpus.

Quanteda assumes you work with corpus objects.

  • can be constructed from individual strings (= character vectors)
  • or from a data.frame (or data.table)

A corpus from character vectors

corpus_1 = corpus(c(text_1, text_2))

A corpus from a data.table

data_1 = fread('/Users/bennettkleinberg/GitHub/snlp/data/real_world_worry_waves_dataset.csv')

names(data_1)[1:13]
##  [1] "id"                   "prolific_id"          "worry_wave1"         
##  [4] "chosen_emotion_wave1" "anger_wave1"          "disgust_wave1"       
##  [7] "fear_wave1"           "anxiety_wave1"        "sadness_wave1"       
## [10] "happiness_wave1"      "relaxation_wave1"     "desire_wave1"        
## [13] "text_long_wave1"

corpus_rwwd = corpus(data_1$text_long_wave1)

corpus_rwwd
## Corpus consisting of 1,152 documents.
## text1 :
## "I am more worried about getting access to my normal healthca..."
## 
## text2 :
## "I am a little concerned that so many people are not staying ..."
## 
## text3 :
## "I feel relaxed. I'm aware of the risks and I'm doing my very..."
## 
## text4 :
## "I feel a little stressed by it all, I would like to go outsi..."
## 
## text5 :
## "I am fed up about not being able to go out when I want, I mi..."
## 
## text6 :
## "At this very moment, I am feeling very fearful and sad about..."
## 
## [ reached max_ndoc ... 1,146 more documents ]

summary(corpus_rwwd, n = 10)
## Corpus consisting of 1152 documents, showing 10 documents:
## 
##    Text Types Tokens Sentences
##   text1    89    132         6
##   text2    70    106         5
##   text3    72    114         6
##   text4    71    108         4
##   text5    71    117         5
##   text6    68    120         6
##   text7    73    118         2
##   text8    65    109         5
##   text9    91    136         3
##  text10    88    124         4

Now: PRACTICAL I

  • becoming comfortable with text handling in R
  • calculating text metrics on real data

Text representations

Aim of text representations

A numerical representation of textual data.

Most common approach:

  • representing a text by its tokens
  • each text consists of a frequency of its tokens

By hand

I think I believe him

How can we construct a term-frequency table?

Steps

  1. create a column for each token
  2. count the frequency
text_id I think believe him
text1 2 1 1 1

Term frequency

  • frequency of each token in each document
  • represented in a table (matrix)
  • tokens are features of a document
  • called a Document Feature Matrix (= DFM)

The DFM in quanteda

  • from tokens or corpus objects, create a DFM table
ex_3 = 'I think I believe him'

corpus_ex_3 = corpus(ex_3)
tokens_ex_3 = tokens(corpus_ex_3)

dfm(tokens_ex_3
    , tolower = T)
## Document-feature matrix of: 1 document, 4 features (0.00% sparse) and 0 docvars.
##        features
## docs    i think believe him
##   text1 2     1       1   1

Sparsity

Sparsity = % of zero-cells

  • Why is the sparsity 0% here?
  • What would you expect if we take additional documents, and why?

DFM with multiple documents

corpus_3 = corpus(c("I think I believe him", "This is a cool function"))
tokens_corpus_3 = tokens(corpus_3)
dfm(tokens_corpus_3)
## Document-feature matrix of: 2 documents, 9 features (50.00% sparse) and 0 docvars.
##        features
## docs    i think believe him this is a cool function
##   text1 2     1       1   1    0  0 0    0        0
##   text2 0     0       0   0    1  1 1    1        1

Remember n-grams?

What happens if we add bi-grams (i.e., \(n=1\) and \(n=2\))?

I think I believe him

ngrams in R

tokens_ngrams(tokens(ex_3), n = 1:2)
## Tokens consisting of 1 document.
## text1 :
## [1] "I"           "think"       "I"           "believe"     "him"        
## [6] "I_think"     "think_I"     "I_believe"   "believe_him"

So we very quickly run into dimensionality problems with text data.

The Real-World Worry Dataset corpus

Unigrams (~ tokens)

tokens_corpus_rwwd = tokens(corpus_rwwd)

dfm_unigrams = dfm(tokens_ngrams(tokens_corpus_rwwd
                                 , n = 1)
                   , tolower = T)

dfm_unigrams
## Document-feature matrix of: 1,152 documents, 6,274 features (98.71% sparse) and 0 docvars.
##        features
## docs    i am more worried about getting access to my normal
##   text1 7  3    1       1     2       1      1  5  5      1
##   text2 3  2    0       1     0       0      0  3  0      0
##   text3 6  0    0       0     2       0      0  5  3      0
##   text4 4  0    0       0     0       0      0  6  1      2
##   text5 7  2    1       1     1       0      0  7  4      0
##   text6 7  3    0       1     4       0      0  2  3      0
## [ reached max_ndoc ... 1,146 more documents, reached max_nfeat ... 6,264 more features ]

Unigrams, bigrams and trigrams

dfm_ngrams = dfm(tokens_ngrams(tokens_corpus_rwwd
                                 , n = 1:3)
                   , tolower = T)

dfm_ngrams
## Document-feature matrix of: 1,152 documents, 150,549 features (99.79% sparse) and 0 docvars.
##        features
## docs    i am more worried about getting access to my normal
##   text1 7  3    1       1     2       1      1  5  5      1
##   text2 3  2    0       1     0       0      0  3  0      0
##   text3 6  0    0       0     2       0      0  5  3      0
##   text4 4  0    0       0     0       0      0  6  1      2
##   text5 7  2    1       1     1       0      0  7  4      0
##   text6 7  3    0       1     4       0      0  2  3      0
## [ reached max_ndoc ... 1,146 more documents, reached max_nfeat ... 150,539 more features ]

dfm_ngrams[1:10, 201:210]
## Document-feature matrix of: 10 documents, 10 features (86.00% sparse) and 0 docvars.
##        features
## docs    a_walk walk_, ,_but but_that that_is is_not not_possible possible_right
##   text1      1      1     1        1       1      1            1              1
##   text2      0      0     0        0       0      0            0              0
##   text3      0      0     0        0       0      0            0              0
##   text4      0      0     0        0       0      1            0              0
##   text5      0      0     0        0       0      0            0              0
##   text6      0      0     0        0       0      0            0              0
##        features
## docs    right_now now_.
##   text1         1     1
##   text2         0     0
##   text3         0     0
##   text4         0     0
##   text5         0     0
##   text6         0     0
## [ reached max_ndoc ... 4 more documents ]

Reducing dimensionality in DFM representations

High-dimensionality is undesirable

  • adds little information
  • statistically problematic
  • computationally inefficient

But: not all words (ngrams) are equally important.

Most common terms

topfeatures(dfm_ngrams, n = 20)
##     .     i   the   and    to     ,   ._i    my    of     a    is    am  that 
##  6464  6049  5842  5234  4369  2966  2344  2289  2091  2035  1928  1775  1769 
##    it  i_am    in   for    as about  this 
##  1715  1669  1628  1567  1429  1410  1346

These add little information.

Also known as stopwords and often removed in preprocessing.

Stopwords

Common stopword lists:

stopwords(language = 'en')[1:20]
##  [1] "i"          "me"         "my"         "myself"     "we"        
##  [6] "our"        "ours"       "ourselves"  "you"        "your"      
## [11] "yours"      "yourself"   "yourselves" "he"         "him"       
## [16] "his"        "himself"    "she"        "her"        "hers"

Note: no universal agreement on stopwords!

Removing stopwords

tokens_rwwd = tokens_select(tokens_corpus_rwwd
                            , pattern = stopwords("en")
                            , selection = "remove")

dfm(tokens_ngrams(tokens_rwwd
                  , n = 1:3)
    , tolower = T)
## Document-feature matrix of: 1,152 documents, 113,954 features (99.85% sparse) and 0 docvars.
##        features
## docs    worried getting access normal healthcare moment , concerned corona .
##   text1       1       1      1      1          1      1 6         1      1 5
##   text2       1       0      0      0          0      0 2         1      0 5
##   text3       0       0      0      0          0      0 3         0      0 6
##   text4       0       0      0      2          0      0 5         0      0 4
##   text5       1       0      0      0          0      0 4         0      0 4
##   text6       1       0      0      0          0      1 2         0      1 6
## [ reached max_ndoc ... 1,146 more documents, reached max_nfeat ... 113,944 more features ]

Special language-based dimensionality reduction

Take these example:

  1. Tokens: creating, creates, create
  2. Tokens: gone, went, going, goes

These all belong to the same “core”.

Stemming and lemmatisation can solve this.

Stemming

Idea: reduce a word to its word stem. Works by removing suffixes.

toks
## Tokens consisting of 1 document.
## text1 :
## [1] "The"       "whole"     "situation" "makes"     "me"        "very"     
## [7] "anxious"   "."
tokens_wordstem(toks)
## Tokens consisting of 1 document.
## text1 :
## [1] "The"     "whole"   "situat"  "make"    "me"      "veri"    "anxious"
## [8] "."

tokens_wordstem(tokens(c('creating creates create')))
## Tokens consisting of 1 document.
## text1 :
## [1] "creat" "creat" "creat"
tokens_wordstem(tokens(c('gone went going goes')))
## Tokens consisting of 1 document.
## text1 :
## [1] "gone" "went" "go"   "goe"

Lemmatisation

Idea: reduce a word to its root meaning (the lemma)

library(textstem)
lemmatize_words(c('gone', 'went', 'going', 'goes'))
## [1] "go" "go" "go" "go"
#lemmatize_words(c('better', 'best'))

Impact of stemming

The RWWD dataset:

  • original ngram representation: 150,549 features (99.79% sparse)
  • after stopword removal: 113,954 features (99.85% sparse)

dfm_stopw = dfm(tokens_ngrams(tokens_rwwd
                  , n = 1:3)
                , tolower = T)
dfm_red = dfm_wordstem(x = dfm_stopw)
dfm_red
## Document-feature matrix of: 1,152 documents, 106,412 features (99.84% sparse) and 0 docvars.
##        features
## docs    worri get access normal healthcar moment , concern corona .
##   text1     1   2      1      1         1      1 6       1      1 5
##   text2     1   1      0      0         0      0 2       1      0 5
##   text3     0   0      0      0         0      0 3       1      0 6
##   text4     0   2      0      2         0      0 5       0      0 4
##   text5     1   1      0      0         0      0 4       0      0 4
##   text6     1   0      0      0         0      1 2       0      1 6
## [ reached max_ndoc ... 1,146 more documents, reached max_nfeat ... 106,402 more features ]

Further reducing sparsity

We still have high sparsity: 106,412 features (99.84% sparse)

Ideas?

Sparsity correction

Trimming the DFM based on term- or document-frequencies:

dfm_red_2 = dfm_trim(dfm_red
                , min_docfreq = 0.05
                , docfreq_type = 'prop')
dfm_red_2
## Document-feature matrix of: 1,152 documents, 260 features (87.51% sparse) and 0 docvars.
##        features
## docs    worri get normal moment , concern corona . health even
##   text1     1   2      1      1 6       1      1 5      1    1
##   text2     1   1      0      0 2       1      0 5      0    0
##   text3     0   0      0      0 3       1      0 6      0    0
##   text4     0   2      2      0 5       0      0 4      0    0
##   text5     1   1      0      0 4       0      0 4      0    0
##   text6     1   0      0      1 2       0      1 6      2    0
## [ reached max_ndoc ... 1,146 more documents, reached max_nfeat ... 250 more features ]

A remaining problem…

We assume all words are equally important.

But:

  • some words occur all by the nature of language (e.g., stopwords)
  • or by the nature of the specific context (here: worries about COVID-19)

We want to:

  • weigh words so that we
    • ‘reward’ local importance (within documents)
    • ‘punish’ for global occurrences (across documents)

Weighting of words

Metric for word importance: frequency of term \(t\) in document \(d\)

  • \(tf(t, d) = \sum_{t \in d} t\)
  • \(tf(t, d) = \frac{\sum_{t \in d} t}{nwords_d}\) (proportion)

Term proportions

dfm_weight(dfm_red_2
           , scheme = 'prop')[1:10, 1:10]
## Document-feature matrix of: 10 documents, 10 features (54.00% sparse) and 0 docvars.
##        features
## docs         worri        get     normal     moment          ,    concern
##   text1 0.02040816 0.04081633 0.02040816 0.02040816 0.12244898 0.02040816
##   text2 0.02222222 0.02222222 0          0          0.04444444 0.02222222
##   text3 0          0          0          0          0.07894737 0.02631579
##   text4 0          0.04081633 0.04081633 0          0.10204082 0         
##   text5 0.02325581 0.02325581 0          0          0.09302326 0         
##   text6 0.01694915 0          0          0.01694915 0.03389831 0         
##        features
## docs        corona          .     health       even
##   text1 0.02040816 0.10204082 0.02040816 0.02040816
##   text2 0          0.11111111 0          0         
##   text3 0          0.15789474 0          0         
##   text4 0          0.08163265 0          0         
##   text5 0          0.09302326 0          0         
##   text6 0.01694915 0.10169492 0.03389831 0         
## [ reached max_ndoc ... 4 more documents ]

Term counts

dfm_weight(dfm_red_2
           , scheme = 'count')[1:10, 1:10]
## Document-feature matrix of: 10 documents, 10 features (54.00% sparse) and 0 docvars.
##        features
## docs    worri get normal moment , concern corona . health even
##   text1     1   2      1      1 6       1      1 5      1    1
##   text2     1   1      0      0 2       1      0 5      0    0
##   text3     0   0      0      0 3       1      0 6      0    0
##   text4     0   2      2      0 5       0      0 4      0    0
##   text5     1   1      0      0 4       0      0 4      0    0
##   text6     1   0      0      1 2       0      1 6      2    0
## [ reached max_ndoc ... 4 more documents ]

Correcting for global occurrence

  • Number of documents \(d\) with term \(t\)
  • \(df(d, t) = \sum_{d \ni t} 1\)
docfreq(dfm_red_2
        , scheme = 'count')[1:10]
##   worri     get  normal  moment       , concern  corona       .  health    even 
##     673     487     216     184     894     213     104    1138     272     146

Simple document frequency

Correcting for term inflation

  • we want: low values for common words
  • and: higher values for important words

Trick: inverse document-frequency

Inverse DF (IDF)

\(idf(t) = \frac{n}{df(d, t)}\) for \(n\) documents in the corpus.

IDF

Problem?

When \(n\) becomes really large…

Solution: log IDF

  • to avoid extreme values, we use the logarithm
  • simple transformation: \(idf(t) = log(\frac{n}{df(d, t)+1})\)

Log IDF

Combining term frequency and inverse document frequency (TF-IDF)

  1. Local importance (TF)
dfm_weight(dfm_red_2, scheme = 'count')[1:10, 1:10]
## Document-feature matrix of: 10 documents, 10 features (54.00% sparse) and 0 docvars.
##        features
## docs    worri get normal moment , concern corona . health even
##   text1     1   2      1      1 6       1      1 5      1    1
##   text2     1   1      0      0 2       1      0 5      0    0
##   text3     0   0      0      0 3       1      0 6      0    0
##   text4     0   2      2      0 5       0      0 4      0    0
##   text5     1   1      0      0 4       0      0 4      0    0
##   text6     1   0      0      1 2       0      1 6      2    0
## [ reached max_ndoc ... 4 more documents ]

  1. Correct for global occurrences (IDF)
docfreq(dfm_red_2
        , scheme = 'inverse'
        , k = 1)[1:10]
##       worri         get      normal      moment           ,     concern 
## 0.232792583 0.373032657 0.724992745 0.794280751 0.109629444 0.731038706 
##      corona           .      health        even 
## 1.040263180 0.004928755 0.625289832 0.894135144

Full flow in R for TF-IDF weighting

dfm_tfidf = dfm_tfidf(dfm_red_2
                      , scheme_tf = "count"
                      , scheme_df = "inverse")

dfm_tfidf
## Document-feature matrix of: 1,152 documents, 260 features (87.51% sparse) and 0 docvars.
##        features
## docs        worri       get    normal    moment         ,   concern   corona
##   text1 0.2334374 0.7478470 0.7269987 0.7966347 0.6606898 0.7330729 1.044419
##   text2 0.2334374 0.3739235 0         0         0.2202299 0.7330729 0       
##   text3 0         0         0         0         0.3303449 0.7330729 0       
##   text4 0         0.7478470 1.4539975 0         0.5505748 0         0       
##   text5 0.2334374 0.3739235 0         0         0.4404598 0         0       
##   text6 0.2334374 0         0         0.7966347 0.2202299 0         1.044419
##        features
## docs             .    health      even
##   text1 0.02655109 0.6268836 0.8970996
##   text2 0.02655109 0         0        
##   text3 0.03186130 0         0        
##   text4 0.02124087 0         0        
##   text5 0.02124087 0         0        
##   text6 0.03186130 1.2537672 0        
## [ reached max_ndoc ... 1,146 more documents, reached max_nfeat ... 250 more features ]

Image reference

Arithmetic operations with text representations

Text similarity

  • each text can be represented as a vector
  • e.g., simple DFM or a TF-IDF weighted DFM

doc_id worri get normal moment make feel anxious
text1 0.2334374 0.7478470 0.7269987 0.7966347 1.2764132 0.3510523 0.4604796
text2 0.2334374 0.3739235 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text3 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.3510523 0.0000000
text4 0.0000000 0.7478470 1.4539975 0.0000000 0.0000000 0.5265784 0.0000000
text5 0.2334374 0.3739235 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text6 0.2334374 0.0000000 0.0000000 0.7966347 0.0000000 0.5265784 0.4604796
text7 0.4668748 0.3739235 0.7269987 0.0000000 0.6382066 0.7021046 0.4604796
text8 0.2334374 0.0000000 0.0000000 0.0000000 0.0000000 0.1755261 0.4604796
text9 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.5265784 0.4604796
text10 0.0000000 0.3739235 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text11 0.4668748 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text12 0.2334374 1.1217706 0.7269987 0.0000000 0.0000000 0.5265784 0.0000000
text13 0.2334374 0.3739235 0.0000000 0.7966347 0.0000000 0.1755261 0.9209592
text14 0.4668748 0.7478470 0.0000000 0.0000000 0.6382066 0.0000000 0.0000000
text15 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text16 0.2334374 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text17 0.0000000 0.7478470 0.0000000 0.7966347 0.0000000 0.0000000 0.0000000
text18 0.2334374 0.0000000 0.0000000 0.0000000 0.0000000 0.3510523 0.0000000
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text1109 0.0000000 0.0000000 0.0000000 0.0000000 0.6382066 0.7021046 0.0000000
text1110 0.2334374 0.0000000 0.7269987 0.7966347 0.6382066 0.0000000 0.9209592
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text1112 0.4668748 0.3739235 0.0000000 0.7966347 0.6382066 0.1755261 0.0000000
text1113 0.0000000 0.0000000 0.0000000 0.7966347 0.0000000 0.0000000 0.0000000
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text1115 0.7003122 0.7478470 0.7269987 0.0000000 0.0000000 0.3510523 0.4604796
text1116 0.2334374 0.0000000 0.0000000 0.0000000 0.0000000 0.3510523 0.9209592
text1117 0.7003122 0.0000000 0.0000000 1.5932693 0.0000000 0.7021046 0.0000000
text1118 0.0000000 1.8696176 0.0000000 0.0000000 0.0000000 0.1755261 0.4604796
text1119 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
text1120 0.2334374 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4604796
text1121 0.4668748 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4604796
text1122 0.2334374 0.3739235 0.0000000 0.0000000 0.0000000 0.3510523 0.0000000
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text1132 0.0000000 0.3739235 0.0000000 0.7966347 0.6382066 0.3510523 0.9209592
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text1136 0.2334374 0.3739235 0.0000000 0.0000000 0.0000000 0.1755261 0.0000000
text1137 0.0000000 0.0000000 0.0000000 0.0000000 0.6382066 0.7021046 0.0000000
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text1141 0.0000000 0.0000000 0.0000000 0.0000000 0.6382066 0.3510523 0.4604796
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text1145 0.9337497 0.0000000 0.0000000 1.5932693 0.6382066 0.1755261 0.4604796
text1146 0.2334374 0.0000000 0.0000000 0.0000000 0.6382066 0.1755261 0.0000000
text1147 0.2334374 0.3739235 0.0000000 0.0000000 0.0000000 0.5265784 0.0000000
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Distances between vectors

Suppose we have got two vectors:

  • \(\vec{v_1} = [1, 2]\)
  • \(\vec{v_2} = [4, 3]\)

Euclidean distance

Uses Pythagorean theorem.

For two 2-dimensional locations:

  • build a right triangle
  • use \(c^2 = a^2 + b^2\) to calculate the length of the hypotenuse \(c\)

By hand:

  • \(a = x_2 - x_1 = 4 - 1 = 3\)
  • \(b = y_2 - y_1 = 3 - 2 = 1\)
  • \(c^2 = a^2 + b^2\)

Thus:

\(c = \sqrt{a^2 + b^2} = \sqrt{9 + 1} = 3.16\)

For \(n\) dimensions:

\(d(\mathbf{x}, \mathbf{y}) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}\)

Euclidean distance and magnitude

  • takes into account the magnitude of vectors
  • but this is not always meaningful
  • different metric: cosine distance
  • bounded to range: -1.00 (opposite) to +1.00 (identical)

Cosine distance

Cosine similarity

\(\text{cosine similarity}(\mathbf{A}, \mathbf{B}) = \frac{A \times B}{\sqrt{A \times A} * \sqrt{B \times B}}\)

Note: \(A \times B\) is the dot product of the vector.

In R

V1 = c(4,2,3)
V2 = c(1,3,1)

cossim = function(A, B){
  numerator = sum(A*B)
  denominator = sqrt(sum(A*A))*sqrt(sum(B*B))
  cosine_sim = numerator/denominator
  return(cosine_sim)
}

cossim(V1, V2)
## [1] 0.7278603

Similarity in our corpus

# Euclidean distance-based
dist(matrix(c(dfm_tfidf_df[1, 2:ncol(dfm_tfidf_df)]
              , dfm_tfidf_df[2, 2:ncol(dfm_tfidf_df)])
            , nrow = 2
            , byrow = T)
     , method = 'euclidean')
##          1
## 2 7.746157

# Cosine distance-based
1 - cossim(A = dfm_tfidf_df[1, 2:ncol(dfm_tfidf_df)]
           , B = dfm_tfidf_df[2, 2:ncol(dfm_tfidf_df)])
## [1] 0.8857499

Now: PRACTICAL II

  • working with text representations
  • preprocessing steps for DFMs
  • vector-based similarity calculations